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Hardy Martingales and Jensen's inequality

Published online by Cambridge University Press:  17 April 2009

Nakhlé H. Asmar  and
Stephen J. Montgomery-Smith
Nakhlé H. Asmar
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, MI 65211, United States of America
Stephen J. Montgomery-Smith
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, MI 65211, United States of America
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Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus 𝕋N, where analyticity is defined using a lexicographic order on the dual group ℤN. We show how, by using basic properties of orders on ℤN, we can apply Garling's method in the study of analytic functions on an arbitrary compact Abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalised Jensen's Inequality due to Helson and Lowdenslager[5].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 2 , April 1997 , pp. 185 - 195
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Asmar, N., ‘The conjugate function on the finite dimensional torus’, Canad. Math. Bull. 32 (1989), 140–148.CrossRefGoogle Scholar
[2]Edwards, R.E. and Gaudry, G.I., Littlewood-Paley and multiplier theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 90 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[3]Garling, D.J.H., ‘On martingales with values in a complex Banach space’, Math. Proc. Cambridge Philos. Soc. 104 (1988), 399–406.CrossRefGoogle Scholar
[4]Garling, D.J.H., ‘Hardy martingales and the unconditional convergence of martingales’, Bull. London Math. Soc. 23 (1991), 190–192.CrossRefGoogle Scholar
[5]Helson, H. and Lowdenslager, D., ‘Prediction theory and Fourier series in several variables’, Acta Math. 99 (1958), 165–202.CrossRefGoogle Scholar
[6]Helson, H. and Lowdenslager, D., ‘Prediction theory and Fourier series in several variables II’, Acta Math. 106 (1961), 175–212.CrossRefGoogle Scholar
[7]Hewitt, E. and Ross, K.A., Abstract harmonic analysis I (2nd edition), Grundlehren der Math. Wissenschaften 115 (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[8]Hewitt, E. and Ross, K.A., Abstract harmonic analysis II, Grundlehren der Math. Wissenschaften in Einzeldarastellungen 152 (Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar
[9]Hewitt, E. and Stromberg, K., Real and abstract analysis, Graduate Texts in Mathematics 25, (2nd priniting) (Springer-Verlag, Berlin, Heidelberg, New York, 1969).Google Scholar
[10]Katznelson, Y., An introduction to harmonic analysis (John Wiley, New York, 1968).Google Scholar